The order dimension of two levels of the Boolean lattices
type: Item
instance of: scholarly article
title: The order dimension of two levels of the Boolean lattices
zbMATH Open document ID: 0810.06004
DOI: 10.1007/BF01462226
published in: Order
publication date: 1994-11-10T00:00:00Z
review text: The Boolean lattice of all subsets of a set having \(n\) elements is denoted by \(B_ n\). If \(0\leq s< t\leq n\), then \(B_ n(s,t)\) is the subset of \(B_ n\) consisting of all elements of \(B_ n\) which are sets with \(s\) or with \(t\) elements; the ordering on \(B_ n(s,t)\) is the restriction of that on \(B_ n\). The dimension of a partially ordered set \(P\) is the minimum positive integer \(d\) such that \(P\) can be embedded into the \(d\)th Cartesian power of the (naturally ordered) set of all real numbers. The dimension of \(B_ n(s,t)\) is denoted by \(\dim(n; s,t)\). The paper gives a survey of results concerning \(\dim(n; s,t)\). Then two inequalities are stated and the main theorem is proved: For \(k\geq 3\) and \(n> 250 k^ 3\), we have \(\dim(n; k,n- k)= n- 2\).
reviewed by: Bohdan Zelinka
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 683158
zbMATH Keywords: suborder
zbMATH Keywords: Boolean lattice
zbMATH Keywords: dimension
zbMATH Keywords: partially ordered set
author: Zoltán Füredi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01462226
OpenAlex ID: W2060891088
description: scientific article; zbMATH DE number 683158